1. Engineering Fundamentals and Problem Solving, 6e Chapter 6 Engineering Measurements

2. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter Objectives Determine the number of significant digits in a measurement Perform numerical computations with measured quantities and express the answer with the appropriate number of significant digits Define accuracy and precision in measurements Define systematic and random errors and explain how they occur in measurements 2

3. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Accuracy and Precision Not Accurate Not Precise Precise but Not Accurate Accurate and Precise Accurate but Not Precise 3

4. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Presentation of Numbers Less than zero: 0.234 not.234 Divide numbers of three orders of magnitude or more with spaces not commas: 1 234.432 1 not 1,234.432,1 Use scientific notation for compactness: 9.87(10) 6 not 9 870 000 4

5. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Use of Prefixes Convenient method of representing measurements 5

6. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Significant Figures Any digit used to express a number, except those zeros used to locate the decimal point. Examples: 0.00123 (3 significant figures) 1.00123 (6 significant figures) 1 000 000 (1 significant figure) 1.000 000 (7 significant figures) 0.100 (3 significant figures) 6

7. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Significant Figures Use scientific notation to clarify significant figures Example: 3 000 (1, 2, 3, or 4 sig. fig?) 3(10 3 ) (1 significant figure) 3.0(10 3 ) (2 significant figures) etc. 7

8. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Measurements Counts (exact values): All digits are significant 32 baseballs (2 sig. fig.) 5 280 ft in a mile (4 sig. fig.) Measured Quantities Measurements are estimates. The number of significant figures depends upon several variables: −instrument graduations, −environment, −reader interpretation, etc. 8

9. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Measurements (con’t) Bar is between 2 and 3 inches Think of it as 2.5 ± 0.5 inches Estimate between 2.6 and 2.7 inches or 2.65 ± 0.05 inches “Best” estimate 2.64 inches with the understanding that the 4 is doubtful 9

10. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Measurements (con’t) Standard practice: In a measurement, count one doubtful digit as significant. Therefore the length of the bar is recorded as 2.64. For calculation purposes the result has 3 significant figures. 10

11. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Arithmetic Operations and Significant Figures General Rule for Rounding To round a value to a specified number of significant figures, increase the last digit retained by 1 if the first figure dropped is 5 or greater. 15.750 becomes 15.8 (3 sig. fig.) 0.015 4 becomes 0.15 (2 sig.fig.) 34.49 becomes 34.5 (3 sig. fig.) or 34 (2 sig. fig.) 11

12. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Arithmetic Operations and Significant Figures General Rule for Multiplication and Division The product or quotient should contain the same number of significant digits as are contained in the number with the fewest significant digits. Examples (15)(233) = 3495 (4 sig. fig. if exact numbers) (15)(233) = 3500 (2 sig. fig. if numbers are measurements) (24 hr/day)(34.33 days) = 823.9 hr (4 sig. fig.) (since 24 is an exact value) 12

13. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Arithmetic Operations and Significant Figures General Rule for Addition and Subtraction The answer should show significant digits only as far to the right as seen in the least precise number in the calculation. Note: last digit in a measurement is doubtful. Example (color indicates doubtful digit) 237.62 28.3 119.743 385.663 By our rules, we keep one doubtful digit. The answer is 385.7 13

14. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Arithmetic Operations and Significant Figures Combined Operations With a calculator or computer, perform the entire calculation and then report result to a reasonable number of significant figures. Common sense application of the rules is necessary to avoid problems. 14

15. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Accounting for Errors in Measurements Measurements can be expressed in 2 parts: A number representing a mean value of the physical quantity measured An amount of doubt (error) in the mean value Example 1: 52.5 ± 0.5 Example 2: 150 ± 2% so 150 means: 147 - 153 The amount of doubt provides the accuracy of the measurement 15

16. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Categories of Error Systematic: Error is consistently in the same direction from the true value. - Errors of instrument calibration - Improper use of measurement device - External effects (e.g. temperature) on measurement device - Must be quantified as much as possible for computation purposes 16

17. Engineering: Fundamentals and Problem Solving, 6e Eide  Jenison  Northup  Mickelson Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. Categories of Error (con’t) Random: Errors fluctuate from one measurement to another for the same instrument. - Measurements usually distributed around the true value - May be caused by sensitivity of instrument - Statistical analysis required 17